However such a scenario does not occur very often in

However, such a scenario does not occur very often. In fact, [Tuc75] has shownthat if no three lightpaths in a given traffic pattern cover the entire ring, then32Lwavelengths are sufficient to perform the wavelength assignment. This is an examplewhere the worst-case nonblocking model results in overdesigning the network. Inorder to support a few pathological patterns, we end up using approximatelyL2additional wavelengths.Let us see what can be gained by having wavelength conversion capabilities ina ring network. If we have full conversion capabilities at all the nodes, then we cansupport all lightpath requests with loadL≤W. However, the same result can beachieved by providing much less conversion capabilities, as shown by the followingresults.Theorem 10.4[RS97]Consider a ring network that has full wavelength con-version at one node and no wavelength conversion at the other nodes. Thisnetwork can support all lightpath requests with loadL≤W.The proof of this result is left as an exercise (Problem 10.19).Limited-wavelength conversion can help significantly in improving the load thatcan be supported in many network configurations. The detailed derivations of theresults for this case are beyond the scope of this book. We summarize the key resultshere.

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614WDM Network DesignFixed-wavelengthconverterWDM mux/demuxFigure 10.20A ring network with fixed-wavelength conversion at one node and no conversionat the others that is able to support lightpath requests with loadL≤W−1. One of the nodes isconfigured to convert wavelengthito wavelength(i+1)modW, and the other nodes provide nowavelength conversion.Theorem 10.5[RS97]Consider the ring network shown in Figure 10.20,which has fixed-wavelength conversion at one node where wavelengthiis con-verted to wavelength(i+1)modW, and no wavelength conversion at the othernodes. This network can support all lightpath requests with loadL≤W−1.By havingd=2limited conversion at two nodes and no conversion at the others,it is possible to improve this result toL≤W[RS97], making such a network asgood as a network with full wavelength conversion at each node.Other topologies such as star networks and tree networks have also been consid-ered in the literature. In star and tree networks,32Lwavelengths are sufficient to doWA-NC [RU94]. In star networks,Lwavelengths are sufficient for WA-FC [RS97].The same result can be extended to arbitrary networks where lightpaths are at mosttwo hops long. Table 10.3 summarizes the results to date on this problem. It is stilla topic of intense research.Multifiber RingsThe wavelength assignment problem in multifiber rings is considered in [LS00]. In amultifiber ring, each pair of adjacent nodes is connected byk >1fiber pairs:k >1fibers are used for each direction of transmission instead of 1 fiber. Recall that we areconsidering undirected edges and lightpaths, and each edge represents a pair of fibers,one for each direction of transmission. Thus, such a multifiber ring is represented by

10.5Maximum Load Dimensioning Models615Table 10.3Number of wavelengths required to perform ofﬂine wavelength as-signment as a function of the loadLwith and without wavelength converters. The

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